Department of Mathematics

UNIVERSITY OF ROCHESTER

Rochester, New York 14627

Tel (716) 275-4429 or 244-9368

FAX (716) 244 6631 (at U of R)

email: RARM@math.rochester.edu

 

Ralph A. Raimi

Professor Emeritus

 

30 August 2000

 

 

To the Editors,

Journal for Research in Mathematics Education:

1906 Association Drive

Reston, VA 20191-9988

 

          Mathematicians Writing, by Leone Burton and Candia Mor­gan, in volume 31 of JRME (July, 2000) pp 429-453, contains some errors that should be noted in JRME, and amended before their interesting research is pushed much further.  I refer in par­ticular to the Indicators of Authority that came under discus­sion on page 439-440.

          "One important aspect of the identities of authors as projected by their texts is the extent to which and the manner in which they claim to be authori­tative within their com­muni­ty.....

          "Terms such as clearly and obvious … [serve] as a claim to authority on the part of the writer (implying that this deriva­tion is clear to me and I do not need to explain it further because if it is not clear to you, that is your fault not mine)­ ...... We believe that, whatever the author's intent, the extent or absence of such words is one of the inter­per­sonal aspects of the writing that will influence the ways in which the reader­s of the text will construct an image of the author and will conse­quently judge the worth of the text it­self."

          Now it might be that -- "what­ever the author's intent" might be -- some readers might judge the character of the author and the worth of his work under the influence of having seen terms such as obvious in that work; but to say -- as Burton and Morgan clearly do in the immediately preceding sentences -- that using such a term con­stitutes a claim to authority is not cor­rect. 

          It is by authority that I know that Napoleon was defeated at Waterloo in 1815, as I have no firsthand knowledge of this alleged fact, only the state­ments (albeit rich and inter­locking) of persons I believe.  We all make use of authority in this way every minute of our lives, and give it little thought. But in the writing of mathe­matical papers,  there is never an appeal to authority. 

          Mathematical statements are almost unique in their failure to appeal to authority. If there is a gap in an ar­gument, the word obvious does the very opposite of con­cealing the gap, and is not asking the reader to accept the wri­ter's word on the matter.  Obvious and clearly merely abbreviate the discus­sion for the benefit of the persons who already know the connecting link that is being omitted. 

          Of course it is a matter of judgment for the author to decide what parts of his argument may be omitted for ex­pository pur­poses, and some authors presup­pose more skill or previous know­ledge on the part of the reader than others do.  Somet­imes the assessment of audience fails; it has often been an occasion of mer­riment among mathe­maticians when someone (in print or at the black­board) imagines obvious that which is not -- not even, on reflec­tion, to the author himself.  But to imagine that the author is expecting his repu­ta­tion to over­whelm the reader by using such phrases for in­timidation is simply mistaken.  There is no form of literature less "authori­tative" than mathema­tics.

        On the following page, a few sentences later, Burton and Morgan write,

          "Indicators of authority claims from the use of modifiers, such as clea­rly, easily, of course, and immediately obvious, and phrases, such as without loss of generality, it suffices to con­sider the case, and the last stage is trivial, all signal a gap in the argument (implying that there is no accom­panying jus­tification for the claim that generality is not lost) and hence signal to the readers that they should accept the author's claim."

          There are many things that can be said about the use of such phrases to abbreviate a mathematical discussion, but that they "signal to the readers that they should a­ccept the author's claim" on the author's authority is not one of the correct ones.

           "Without loss of generality", for example, is a tech­nical phrase used in consideration for the reader (and publisher).  To con­sider it an assertion of authority rather than a part of the proof is analo­gous to considering "Let x be an element of G" an intru­sion on the reader's liberty to consider x in other ways; no mathematician would imagine that the apparently peremptory tone of "Let x be an element of G" constitutes a claim to au­thority.

          An explanation (by example) of without loss of generali­ty might be of value here, for it would surely not be wise to ask the present reader to accept my preceding paragraph on my own authority, either:  In proving by use of a coordinate system the theorem that the medians of a triangle meet at a point, a mathe­matician might begin by saying, "Let ABC be the triangle.  Without loss of generality, suppose A is at the origin (0,0) and AB is along the x-axis with B at the point (b,0)." 

          It is thus asserted, in other words, that prov­ing the theorem for this specially placed triangle proves it for all other triangles.  Had the proof begun by asking, in addition, that the point C be given the coordinates (b,b), the mathemati­cian could be called to order for inap­propri­ate use of "WOLOG" (a common informal abbreviation for without loss of generality, so frequent and so little idiosyncratic is the use of the term), for the triangle thus specified would a right triangle, and proving things for a right triangle is not proving it for all triangles.

          Why, on the contrary, does placing two of the vertices at (0,0) and (0,b) fail to lose generality?  Surely not every tri­angle in the plane oc­cupies so particular a position.  Isn't this mathematician cheating a bit, making it easier for himself by this ploy, intimidating his less secure readers by some claim to authority? 

          Well, it is the genius of the entire method of coor­di­nates, that the Euclidean properties of Eu­clidean figures remain in­variant under rigid motions, that concurrence and distances are Euclidean properties in this sense, and that there is always a rigid motion which will bring any triangle, wher­ever placed initially, to the posi­tion described in this proof. 

          Now the whole story of how Euclidean geometry can be got at via coordinate systems is really quite long  Not everyone has learned these things, and a high school student using analytic geometry for the first time often does take such missing links on faith, sometimes not even noticing that anything is missing.  If the mathe­matician were writing for that high school student he might well avoid "WOLOG" and provide a longer explanation.  Yet mathe­ma­tical and scien­tific ex­position would come to a stop if all papers began "at the begin­ning."  Mathematical research papers, especially, are father from first principles than most expositions because directed at profes­sional mathematicians.  

          On the other hand, the path back, from where the ex­positor is now, to where the first prin­ciples are to be found, must always be visible.  In tack­ling this theorem about the centroid of a triangle my imagined author is pre­suming the reader has already come a certain distance in analytic geome­try.  Yet rather than merely placing the tri­angle's base on the x-axis with a vertex at the origin, this particular ex­position makes a point of reminding the reader, via "WOLOG", that there is no sleight-of-hand at work.

          Far from being an appeal to authori­ty, the phrase without loss of genera­lity is a reminder that some "first principles," or some of the inter­vening logical paths, are being omitted, so that the uncomprehending reader, who might otherwise think him­self obtuse for not seeing why the proof for the spe­cial case proves them all, is given the more pleasant infor­mation that he is merely missing some information.   At the same time, the comprehen­ding read­er, the author's expected audience, is saved some tedium.  So with the other phrases cited by Burton and Morgan as assertions of authority.  Take it from me, they are no such thing.

 

                                                Sincerely yours,

 

                                                Ralph A. Raimi